Extension of Cauchy’s Theorem
Cauchy-Goursat Theorem is an extension of Cauchy’s Integral Theorem for simply connected regions in complex analysis. It states that if a function f(z) is analytic within a simply connected region D and its contour, then the integral of f(z) around any closed contour C within D is zero.
Mathematically, the Cauchy-Goursat Theorem can be stated as follows:
∮C f (z) dz = 0
Where:
- ∮C denotes the contour integral around the closed contour C
- f(z) is an analytic function within the simply connected region D and its contour
Cauchy Theorem
the Cauchy-GoursatCauchy’s Theorem states that if a function is analytic within a closed contour and its interior, then the integral of that function around the contour is zero. This fundamental result is central to complex analysis, providing insights into the behavior of analytic functions in complex domains.
In this article, we will learn about Cauchy’s Integral Theorem, Cauchy’s Integral Formula, It’s applications, Cauchy’s Residue Theorem and the Cauchy-Goursat Theorem.
Table of Content
- What is Cauchy’s Integral Theorem?
- Cauchy’s Integral Formula
- Applications of Cauchy Theorem
- Cauchy’s Residue Theorem
- Extension of Cauchy’s Theorem